Abstract We discuss the three-body decay mechanisms of many-body resonances. R-matrix sequential description is compared with full Faddeev computation. The role of the angular momentum and boson symmetries is also studied. As an illustration we show the computed α-particle energy distribution after the decay of 12 C(1 + ) resonance at 12.7 MeV

R Álvarez-Rodríguez

· A S Jensen

· D V Fedorov

· E Garrido

· H O U Fynbo

· O S Kirsebom

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Few-Body Syst (2009) 45: 149–152
DOI 10.1007/s00601-009-0010-2
R. Álvarez-Rodríguez · A. S. Jensen · E. Garrido ·
D. V. Fedorov · H. O. U. Fynbo · O. S. Kirsebom
Three-Body Decays: Structure, Decay Mechanism
and Fragment Properties
Received: 28 November 2008 / Accepted: 19 January 2009 / Published online: 11 February 2009
© Springer-Verlag 2009
Abstract We discuss the three-body decay mechanisms of many-body resonances. R-matrix sequential
description is compared with full Faddeev computation. The role of the angular momentum and boson sym-
metries is also studied. As an illustration we show the computed α-particle energy distribution after the decay
of 12C(1+) resonance at 12.7 MeV.
1 Introduction
The three-body decay of many-body resonances can be accurately measured in complete kinematics. Informa-
tion about the decaying state and the decay mechanism is usually extracted from the measurement of the three
fragments after the decay. Although this is a common practice, the situation is ambiguous. The experimental
analyses of these processes are based on the R-matrix formalism which inherently assumes two successive
two-body decays. The input are the properties of the intermediate two-body states and the population of the
nuclear many-body initial state approximated as a three-body system.
Occasionally, in principle contrary descriptions are able to explain the observed distributions making the
understanding of the underlying physics difficult. An example demonstrating the difficulties is the 3α decay of
1+ state in 12C which was successfully described by two opposite mechanisms: a sequential decay via the 2+
state in 8Be [1], and a direct decay into the three-body continuum [2]. This requires an explanation. What infor-
mation is contained in a full-kinematics measurement of three-body decay? Apparently unique information
can only be extracted under favorable conditions. The crux of the matter is that the decay mechanism is related
to a “decay path”, an intermediate structure, which in contrast to the final state signal in the detector is not an
observable. We shall in this contribution compare the results from three-body calculations and experimental
R-matrix analyses.
This article is based on the presentation by R. Álvarez-Rodríguez at the Fifth Workshop on Critical Stability, Erice, Sicily.
R. Álvarez-Rodríguez · A. S. Jensen · D. V. Fedorov · H. O. U. Fynbo · O. S. Kirsebom
Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark
E. Garrido
Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
Present address:
R. Álvarez-Rodríguez (B)
INFN Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
E-mail: raquel.alvarez@pi.infn.it
150 R. Álvarez-Rodríguez et al.
2 Energy Distributions
The large-distance observable structure of the many-body initial state is a three-body continuum state, therefore
we compute the resonance structure in a three-body cluster model [3]. We use the complex scaled hyperspherical
adiabatic expansion method to solve the Faddeev equations which describe the 3-body system. The appropriate
coordinates are the so-called hyperspherical coordinates and consist of the hyperradius ρ2 = 4 ∑3i=1(ri −R)2 ,
and five hyperangles. In the adiabatic hyperspherical expansion method the angular part of the Faddeev equa-
tions is solved first and the angular eigenfunctions n J M are then used as a basis to expand the total wave
function  J M .
We include short-range [4] and Coulomb potentials. The many-body effects that are present at short dis-
tances are assumed to be unimportant except for the resonance energy. This is taken into account by using a
structureless 3-body interaction that fits the position of the resonance.
The resonance wave-function contains information about the decay mechanism, and the large-distance
properties reflect directly the measurable fragment momentum distributions. The single particle probability
distributions are obtained after integration of the absolute square of the wave function over the four hyperangles
describing the directions of the momenta.
The many-body initial state resonance evolves into three clusters at large distances. The total angular
momentum and parity Jπ is conserved in the process. This symmetry combined with Bose-statistics imposes
constraints on the resulting momentum distributions. An early example of these effects applied to three pion
decays can be found in [5]. Figure 1 shows the regions of the Dalitz plot where the density must vanish for
the decay of a 1+ state into three α-particles. This gives rise to the minima in the single α-particle energy
distribution. The Dalitz plot computed within the Faddeev framework is also shown and is in agreement with
these symmetry constraints.
Figure 2 shows the single-α energy distributions, i.e. the probability for emergence of one α particle with
a given energy divided by its maximum allowed, for the 1+ state of 12C computed with R-matrix analysis
[1,6]. This corresponds to the projection of the Dalitz plot in Fig. 1 on the y-axis. The decay is assumed to be
sequential via 8Be(2+) since angular momentum forbids the decay via 8Be(0+). We have varied the two-body
energy and width. When both the two-body energy and width are small (left) a narrow peak corresponding
to the emission of the first α arises. The other two α’s are related to the broad peaks. By increasing the two-
body energy and width the three-peak distribution becomes rather pronounced and insensitive to the two-body
parameters when either E2r/E3r is larger than about 0.5 or the two-body width is large. The same figure
contains the curves corresponding to the case where the boson symmetry is omitted. For a low and narrow
two-body state the effect of this symmetry seems to be unimportant, but an increase on the width leads to a
two-peak (not three-peak) distribution.
Figure 3 shows the results from the full three-body computation and the computation from the lowest con-
tinuum three-body wave function (K = 8) from [7,8] (democratic decay). This is the simplest assumption with
the correct symmetries. We have varied the three-body energy and consequently the three-body width. Two
rotation angles have been considered: one of them is large enough to accumulate the contribution of sequential
decay through the 8Be resonance in a single adiabatic potential, while the other is not. The calculations include
the boson symmetry of the α-particles. The results from the large rotation angle do not include the contribution
from the decay via 8Be(2+) and are very close to the democratic decay. In the result from the full Faddeev
computation the three peaks are closer to each other and this approaches better the experiment. The fractions
of population at large distance are given in Table 1 for the different values of E3r shown in Fig. 3. We can
observe that the sequential decay probability increases as we increase the three-body energy.
 1  1.5  2  2.5  3  3.5  4  4.5  5
 0
 0.5
 1
 1.5
 2
 2.5
 3
 3.5
 4
Fig. 1 Regions of the 3α Dalitz plot where the density must vanish (in black) for a 1+ state (left). Dalitz plot for the 1+ state of
12C. x-axis corresponds to (Eα1/2Eα2)/
√
(3) and y-axis to Eα1 in MeV (Color online)
Three-Body Decays: Structure, Decay Mechanism and Fragment Properties 151
0 0.25 0.5 0.75 1
Eα/Eα,max
0
1
2
3
4
5
P(
 E
α/
E
α,
m
ax
 )
 (
ar
b.
 u
ni
ts
)
0 0.25 0.5 0.75 1
Eα/Eα,max
0
1
2
3
0 0.25 0.5 0.75 1
Eα/Eα,max
0
1
2
3
E
2
/E
3
=0.45 E2/E3=0.56 E2/E3=1.1Γ2/E3=0.1 Γ2/E3=0.28 Γ2/E3=0.42
Fig. 2 Single α energy distributions from R-matrix analysis for the decay into three α-particles of the 12C(1+) resonance at
12.7 MeV of excitation energy. The energies and widths of the intermediate 8Be(2+) state are varied as specified in the panels.
The green dashed curve corresponds to the case where the symmetrization of the wave function is omitted. The middle panel
corresponds to the measured resonance energy (Color online)
0 0.25 0.5 0.75 1
Eα/Eα,max
0
1
2
3
P(
 E
α/
E
α,
m
ax
 )
 (
ar
b.
 u
ni
ts
)
0 0.25 0.5 0.75 1
Eα/Eα,max
0
1
2
3
0 0.25 0.5 0.75 1
Eα/Eα,max
0
1
2
3
E
2
/E
3
=0.39 E2/E3=0.56 E2/E3=0.86
Γ
2
/E
3
=0.19 Γ
2
/E
3
=0.28 Γ
2
/E
3
=0.43
Fig. 3 Single α energy distributions from Faddeev computation for direct decay of the 12C(1+) resonance. The three-body energy
is varied by changing the strength of the three-body potential. The relative energies and widths of the intermediate 8Be(2+)
resonance are specified in the panels. The solid (black) and dashed (green) curves correspond to complex rotation angles of
θ = 0.25 and 0.1 respectively. The dotted (blue) curve corresponds to democratic decay [7,8]. For θ = 0.25 only direct decay is
shown (Color online)
Table 1 The probability Pseq for populating the component related to the decay via 8Be(2+) at large distances in the computation
of the 1+ resonance of 12C for a complex rotation angle θ = 0.25
12C(Jπ ) E2r /E3r 2r /E3r E3r (MeV) 3r (MeV) ly Pseq
0.86 0.43 3.5 0.005 2 0.001
1+ 0.56 0.28 5.4 0.09 2 0.12
0.39 0.19 7.8 1.15 2 0.89
The three-body energy (E3r ) is varied by adjusting the strength of the three-body potential. The 8Be(2+) two-body energy is
maintained E2r = 2.7 MeV. The energies are referred to the 3α or 2α separation threshold
3 Conclusions
We have computed the observable momentum distributions from decay of three-body resonances by use of
R-matrix simulations and from full Faddeev calculations. We have considered the example of the 1+ reso-
nance of 12C. The angular momentum and boson symmetries constrain the resulting momentum distributions.
The same measured momentum distributions can be described in different complete basis sets, e.g. either
direct products of two-body states and their center-of-mass motion relative to the third particle (R-matrix) or
three-body continuum wave functions (Faddeev). The fact that different descriptions seem to work indicates
that the same wave function could be described in different ways. Extracting information of both structure
and decay mechanism can then be misleading and requires model interpretations. Full Faddeev computations
successfully reproduce the measured distributions.
152 R. Álvarez-Rodríguez et al.
References
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2. Álvarez-Rodríguez, R., et al.: Energy distributions from three-body decaying many-body resonances. Phys. Rev. Lett. 99,
072503 (2007)
3. Nielsen, E., Fedorov, D.V., Jensen, A.S., Garrido, E.: The three-body problem with short-range interactions. Phys. Rep. 347,
373 (2001)
4. Ali, S., Bodmer, A.R.: Phenomenological α–α potentials. Nucl. Phys. 80, 99 (1966)
5. Zemach, Z.: Three-pion decays of unstable particles. Phys. Rev. 133, B1201 (1964)
6. Fynbo, H.O.U., et al.: Three-body decays and R-matrix analyses (in preparation)
7. Korsheninnikov, A.A.: Yad. Fiz. 52, 1034 (1990)
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Three-Body Decays: Structure, Decay Mechanism and Fragment Properties

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